Undergraduate Texts in Mathematics

Transcription

1 Undergraduate Texts in Mathematics Editors S. Axler F.W. Gehring K.A. Ribet

2 Springer Books on Elementary Mathematics by Serge Lang MATH! Encounters with High School Students 1985, ISBN The Beauty of Doing Mathematics 1985, ISBN Geometry. A High School Course (with G. Murrow), Second Edition 1989, ISBN Basic Mathematics 1988, ISBN A First Course in Calculus 1986, ISBN Calculus of Several Variables 1987, ISBN Introduction to Linear Algebra 1986, ISBN Linear Algebra 1987, ISBN Undergraduate Algebra, Third Edition 2005, ISBN Undergraduate Analysis 1983, ISBN Complex Analysis 1985, ISBN Math Talks for Undergraduates 1999, ISBN

3 Serge Lang Undergraduate Algebra Third Edition Sprin g er

4 Serge Lang Department of Mathematics Yale University New Haven, CT USA Editorial Board S. Axler F.W. Gehring K.A. Ribet Mathematics Department Mathematics Department Mathematics Department San Francisco State East Hall University of California, University University of Michigan Berkeley San Francisco, CA Ann Arbor, MI Berkeley, CA USA USA USA Mathematics Subject Classification (2000); 13-01, Library of Congress Cataloging-in-Publication Data Lang, Serge, Undergraduate algebra / Serge Lang. 3rd ed. p. cm. (Undergraduate texts in mathematics) Includes bibliographical references and index. ISBN (alk. paper) 1. Algebra. I. Title. II. Series. QA152.3.L dc ISBN Printed on acid-free paper. 2005, 1990, 1987 Springer Science-I-Business Media, Inc. All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the pubusher (Springer Science-f Business Media, Inc., 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed in the United States of America. (EB/ASCO) SPIN springeronline.com

5 Foreword This book, together with Linear Algebra, constitutes a curriculum for an algebra program addressed to undergraduates. The separation of the hnear algebra from the other basic algebraic structures fits all existing tendencies affecting undergraduate teaching, and I agree with these tendencies. I have made the present book self contained logically, but it is probably better if students take the linear algebra course before being introduced to the more abstract notions of groups, rings, and fields, and the systematic development of their basic abstract properties. There is of course a little overlap with the book Linear Algebra, since I wanted to make the present book self contained. I define vector spaces, matrices, and linear maps and prove their basic properties. The present book could be used for a one-term course, or a year's course, possibly combining it with Linear Algebra. I think it is important to do the field theory and the Galois theory, more important, say, than to do much more group theory than we have done here. There is a chapter on finite fields, which exhibit both features from general field theory, and special features due to characteristic p. Such fields have become important in coding theory. There is also a chapter on some of the group-theoretic features of matrix groups. Courses in hnear algebra usually concentrate on the structure theorems, quadratic forms, Jordan form, etc. and do not have the time to mention, let alone emphasize, the group-theoretic aspects of matrix groups. I find that the basic algebra course is a good place to introduce students to such examples, which mix abstract group theory with matrix theory. The groups of matrices provide concrete examples for the more abstract properties of groups fisted in Chapter II.

6 VI FOREWORD The construction of the real numbers by Cauchy sequences and null sequences has no standard place in the curriculum, depending as it does on mixed ideas from algebra and analysis. Again, I think it belongs in a basic algebra text. It illustrates considerations having to do with rings, and also with ordering and absolute values. The notion of completion is partly algebraic and partly analytic. Cauchy sequences occur in mathematics courses on analysis (integration theory for instance), and also number theory as in the theory of p-adic numbers or Galois groups. For a year's course, I would also regard it as appropriate to introduce students to the general language currently in use in mathematics concerning sets and mappings, up to and including Zorn's lemma. In this spirit, I have included a chapter on sets and cardinal numbers which is much more extensive than is the custom. One reason is that the statements proved here are not easy to find in the literature, disjoint from highly technical books on set theory. Thus Chapter X will provide attractive extra material if time is available. This part of the book, together with the Appendix, and the construction of the real and complex numbers, also can be viewed as a short course on the naive foundations of the basic mathematical objects. If all these topics are covered, then there is enough material for a year's course. Different instructors will choose different combinations according to their tastes. For a one-term course, I would find it appropriate to cover the book up to the chapter on field theory, or the matrix groups. Finite fields can be treated as optional. Elementary introductory texts in mathematics, like the present one, should be simple and always provide concrete examples together with the development of the abstractions (which explains using the real and complex numbers as examples before they are treated logically in the text). The desire to avoid encyclopedic proportions, and specialized emphasis, and to keep the book short explains the omission of some theorems which some teachers will miss and may want to include in the course. Exceptionally talented students can always take more advanced classes, and for them one can use the more comprehensive advanced texts which are easily available. New Haven, Connecticut, 1987 S. LANG Acknowledgments I thank Ron Infante and James Propp for assistance with the proofreading, suggestions, and corrections. S.L.

7 Foreword to the Third Edition In this new edition I have added new material in Chapters IV and VI, first on polynomials, and second on linear algebra in combination with group theory. The additions to Chapter VI describe various product structures for SL (Iwasawa and other decompositions). These also have to do with the conjugation action and the decomposition of the Lie algebra under this action. The algebra involved comes from deeper theories, but the parts I have extracted on SL belong to an elementary level. Students are then put into contact with some algebra used as a backdrop for analysis on groups, starting with SL. A new section in Chapter IV gives a complete account of the Mason- Stothers theorem about polynomials, with Noah Snyder's beautifully simple proof It is worth emphasizing that the derivative for polynomials is a purely algebraic operation, for which limits are not required. A Springer pamphlet has been pubhshed to present a self-contained treatment of polynomials (from scratch) culminating with this topic. Here it takes its place as a section in the general chapter on polynomials. It occurs as a natural twin for the section on the abc conjecture. I have tried on several occasions to put students in contact with genuine research mathematics, by selecting instances of conjectures which can be formulated in language at the level of this course. I have stated more than half a dozen such conjectures. Of which the abc conjecture provides one spectacular example. Usually students have to wait years before they realize that mathematics is a live activity, sustained by its open problems. I have found it very effective to break down this obstacle whenever possible.

8 Vm FOREWORD TO THE THIRD EDITION Acknowledgment I thank Keith Conrad for his suggestions and help with the proofreading in previous editions. I also thank Allen Altman for numerous additional corrections. New Haven 2004 SERGE LANG

9 Contents Foreword Foreword to the Third Edition v vii CHAPTER I The Integers 1 1. Terminology of Sets 1 2. Basic Properties 2 3. Greatest Common Divisor 5 4. Unique Factorization 7 5. Equivalence Relations and Congruences 12 CHAPTER II Groups Groups and Examples Mappings Homomorphisms Cosets and Normal Subgroups Application to Cyclic Groups Permutation Groups Finite Abelian Groups Operation of a Group on a Set Sylow Subgroups 79 CHAPTER III Rings Rings Ideals Homomorphisms Quotient Fields 100

10 X CONTENTS CHAPTER IV Polynomials Polynomials and Polynomial Functions Greatest Common Divisor Unique Factorization Partial Fractions Polynomials Over Rings and Over the Integers Principal Rings and Factorial Rings Polynomials in Several Variables Symmetric Polynomials The Mason-Stothers Theorem The abc Conjecture 171 CHAPTER V Vector Spaces and Modules Vector Spaces and Bases Dimension of a Vector Space Matrices and Linear Maps Modules Factor Modules Free Abelian Groups Modules over Principal Rings Eigenvectors and Eigenvalues Polynomials of Matrices and Linear Maps 220 CHAPTER VI Some Linear Groups The General Linear Group Structure of Gh^(F) SL,(F) SL (R) and SL (C) Iwasawa Decompositions Other Decompositions The Conjugation Action 254 CHAPTER VII Field Theory Algebraic Extensions Embeddings Splitting Fields Galois Theory Quadratic and Cubic Extensions Solvability by Radicals Infinite Extensions 302 CHAPTER VIM Finite Fields General Structure The Frobenius Automorphism 313

11 CONTENTS XI 3. The Primitive Elements Splitting Field and Algebraic Closure Irreducibility of the Cyclotomic Polynomials Over Q Where Does It All Go? Or Rather, Where Does Some of It Go? CHAPTER IX The Real and Complex Numbers Ordering of Rings Preliminaries Construction of the Real Numbers Decimal Expansions The Complex Numbers 346 CHAPTER X Sets More Terminology Zorn's Lemma Cardinal Numbers Well-ordering 369 Appendix 1. The Natural Numbers The Integers Infinite Sets 379 Index 381